Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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4answers
122 views

How many digits are there in 100!? [on hold]

How many digits are there in 100 factorial? How does one calculate the number of digits?
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1answer
33 views

Number combinations - website [on hold]

There are four directions: up, down, left, right. (I am programming something). Four patterns with one (up, down, left, right), 16 patterns with two (up up, up down, up left, up right) and so on. I ...
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1answer
34 views

how $\prod\limits_{i=1}^{n} (2k-1)/2= (2n)!/{(4^n)n!} = (2n-1)!/[{2^{2n-1}}(n-1)!]$? [on hold]

We know $Γ(n+1/2)=(n-1/2)!= Π (n-1/2)= √π \cdot \prod\limits_{i=1}^{n} (2k-1)/2$ hence $Γ(n+1/2) = (2n)!/{(4^n)n!}√π = (2n-1)!/[2^{2n-1}(n-1)!]\sqrtπ$ But I need the answer of above question to prove ...
3
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3answers
71 views

Writing number as sum of reciprocals of factorial

Given a real number $r>0$. Is there a way to determine whether $r$ can be written as a (possibly infinite) sum of distinct terms of the form $1/n!$? For example, if we want to determine whether ...
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2answers
38 views

Factorial inequality $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n$ using induction

I want to show $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n.$ Assume that it holds for some positive integer $k\geq 1$ and we will prove, $2!\,4!\,6!\cdots (2k+2)!\geq\left((k+2)!\right)^{k+1}$. ...
11
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1answer
277 views

Help with difficult telescoping series question

Evaluate $$\frac3{1!+2!+3!}+\frac4{2!+3!+4!}+\ldots+\frac{2012}{2010!+2011!+2012!}\;.$$ I see that the question is telescoping, but I don't know how to break it down into a form similar to that of ...
0
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1answer
23 views

Is $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} {(\delta+1)^n}$ for any $n$ ?(in this specific case)

Let $\alpha=(K-1)a$, $\beta=K$ and $\delta=Ka$, where $K>a\ge 1$ ($\delta>\alpha>\beta$). Can we claim that $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} ...
1
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3answers
30 views

What is the growth rate of the logarithm of the factorial sequence?

I'd like to know the space complexity of storing bit string representations of the numbers in the factorial sequence (as in a memoized factorial function). So each number $f_i=i!$ in $i=0 \cdots n$ ...
2
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2answers
91 views

Summing of factorials to produce perfect cubes

I was playing around with factorials the other day, and I realized that $4!+5!$ is a perfect square. Perplexed by this result, I started looking for other pairs of factorials that produce a perfect ...
3
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0answers
65 views

Are there infinite primes of the form n!+1? [duplicate]

Are there infinite primes of the form n!+1 ? I searched the internet but couldn't find the answer.
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3answers
32 views

The only solution of the equation ${72_8!}/{18_2!}=4^x$ is $x=9$

Problem and Definitions If $n_a!:=n(n-a)(n-2a)(n-3a)\ldots(n-ka):n>ka$, how should I go about solving this?: $$\dfrac{72_8!}{18_2!}=4^x$$ Attempt ...
3
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4answers
187 views

Formula for $1! \times 2! \times \cdots \times n!$?

Are there any useful forms for the expression $1!\cdot 2!\cdot 3!\cdot ...\cdot n!$? I'm trying to solve a problem that involves this expression and thought it might help to find a more "workable" ...
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2answers
32 views

Conditional Probability in Poker

I'm thinking of a ten person Texas hold'em game. Each person is dealt 2 cards at the start of the game. The question is: GIVEN that you have been dealt 2 hearts (Event B), what is the probability ...
1
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1answer
39 views

What non-integer number has the smallest factorial? [duplicate]

Quick google search for factorial of non-integers led me to gamma function. I tried that in my calculator and it worked as expected for non-integers. Perhaps implements gamma function internally. ...
1
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2answers
31 views

c(n,k) equals subdivisions

To compare n files, the total comparison count is: $$ {{n}\choose{k}} = C^k_n = \dfrac{n!}{k! ( n - k )!} $$ with k = 2. Input space is composed by all pairs of files to compare. I want to split ...
3
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7answers
140 views

For what $n$ is $n! = 2^8\cdot3^4\cdot5^2\cdot7$?

How can one find $n$ when $n! = 2^8\cdot3^4\cdot5^2\cdot7$? And generally, How to solve this kind of questions? The textbook provided a poor answer.
0
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1answer
28 views

How do I prove this by induction? [duplicate]

thank you for taking the time to help me with the question. I am struggling to use proof by induction for this formula: $$\sum_{k=0}^{n}k\times k! = (n + 1)! - 1$$ So far, I came up with: $$S(n) = ...
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5answers
145 views

Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
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0answers
27 views

Additive and Multiplicative Error in $n!$ Approximation

Let $S(n)=\sqrt{2\pi n}\big(\frac{n}{e}\big)^n$ be the approximation of interest to $n!$. What are good lower and upper bounds on the following two functions $$(1)\mbox{ }|S(n)-n!|?$$ $$(2)\mbox{ ...
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2answers
32 views

Ratio of 2 Sums of products of binomial coefficients

I want to prove that for $k \ even, 0 \leq k<n, n\in \mathbb{N}$: $-\frac{1}{(2n-3-k)(k+2)}\sum \limits_{i=0}^{k} \frac{(-1)^{i} 2^{i} (2n-2-i)!}{(n-1-i)!i!(k-i)!}=\sum \limits_{i=0}^{k+2} ...
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0answers
18 views

is there anyone able to develop this series in order to get the following equality?

$\sum_{i=1}^\infty (1-\alpha)_{(i-1)}*\frac{\varepsilon^i}{i!}$ = $\frac{1-(1-\varepsilon)^{\alpha}}{\alpha}$ where $(1-\alpha)_{(i-1)}$ is the Pochammer symbol or rising\ascending factorial. Can ...
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3answers
78 views

clarification on the formula $\frac{n!}{(n-k)!}$

$\dfrac{n!}{(n-k)!}$ is used in order to find non-repetitive lists of length $k$ given $n$ possible symbols. For example: find the number of non-repetitive lists of length five that can be made ...
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1answer
22 views

Inequality between factorial and exponential

Trying to find a nice way to simplify the question: Which is bigger 2000! or 1000^2000? I don't know what kind of reasoning I can apply here.
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1answer
42 views

Find the GCD and LCM of the factorials of two given numbers

Find $\gcd(20!, 12!)$ and $\text{lcm}(20!, 12!)$. My answer is: $20=2^2 \times 5$ $12=2^2 \times 3$ GCD $= 2^2 = 4$ LCM $= 2^2 \times 3 \times 5 = 60$ .... But my teacher said that this symbol ...
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0answers
30 views

Determine the formula for hexagon arrangements.

The puzzle to be solved is similar to a jigsaw but using n regular hexagons of equal size for pieces. The pieces are to be placed within a defined perimeter to create a picture. Q: If we let the ...
1
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2answers
40 views

Factorial simplificaton involving negative 1

What is the best way of simplifying $$\dfrac{(a+b-1)!}{(b-1)!}$$ Ideally i want to get rid of the two $-1$ and the final solution should not containt the gamma function
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1answer
28 views

How to prove that nth differences of a sequence of nth powers would be a sequence of n!

Given an infinite sequence of numbers, first differences denote a sequence of numbers that are pairwise differences, second differences denote a new sequence of pairwise differences of this sequence, ...
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0answers
21 views

How to find a distinct digits appearance from a factorial number?

I want to find out how many times a single digit is appeared in a factorial number. For example- 9! = 362880. There are two times the digit 8 appeared. Again, 13! = 6227020800. Here the digit 2 ...
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1answer
35 views

Bell number vs Factotial

We have $B_n$ is Bell number and $n!$ - factorial. So, what is greater: $n!$ or $B_n$ ? How it can be proven?
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1answer
22 views

Limit of factorials

I'm failing go figure out how to calculate the limit where I have one factorial divided by two at about half its size. The specific limit I'm trying to find is this: $$\lim_{n\to \infty}\frac ...
2
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0answers
85 views

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$, where $\left(\binom{a}{b}\right)=\dfrac{a!!}{b!!(a-b)!!}$ EDIT : Someone pointed out in the Mathematics chat that my ...
1
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2answers
53 views

for all positive integers m there exists consecutive primes which are at least m apart

I'm having difficulty as to how I should approach this problem, any help would me much appreciated! Note that $k$ divides $n! + k$ for each $k\le n$. Use this fact to show that for all positive ...
1
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1answer
25 views

Prove that if $p\le n$, then $p$ does not divide $n! + 1$

I'm having trouble on how to approach this problem Prove that if $p\le n$, then $p$ does not divide $n! + 1$ ($p$ is prime and $n$ is an integer).
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0answers
35 views

Inequality involving factorial and a number 1/12

How I can prove the following two inequalities: If $n$ is a positive integer then $$ \sqrt{2 \pi}n^{n+\frac{1}{2}}e^{-n+\frac{1}{12n+1}}<n!<\sqrt{2 \pi}n^{n+\frac{1}{2}}e^{-n+\frac{1}{12n}} $$
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3answers
168 views

What is the practical application of factorials

I'm trying to understand the practical application of factorial - in simple applications. I searched the math.stackexchange and could not find an answer. I understand that a factorial of n items ...
0
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4answers
94 views

Why is ${{n+1}\choose{k}}={{n}\choose{k-1}}+{{n}\choose{k}}$? [duplicate]

My teacher showed us a proof by induction for this equation for $n\in\mathbb{N}$: $$\sum\limits_{k=0}^n{{n}\choose{k}} = 2^n$$ In the first step, this sum is rewritten using ...
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0answers
41 views

How come negative factorials never give us an answer?

I've done this and it always gave me an error probably because of this (it'll continue):$$4!=4*3*2*1=24$$$$3!={4!\over 4}={24\over 4}=6$$$$2!={3!\over 3}={6\over 3}=2$$$$1!={2!\over 2}={2\over ...
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1answer
16 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
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4answers
64 views

How can we find factorials in decimal form? [duplicate]

I've heard of factorials such as $5!$ and $3!$, which work like this: $5!=5\times4\times3\times2\times1=120$ and $3!=3\times2\times1=6$. At least this is what we get. Also, surprisingly, $0!=1$, but ...
2
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3answers
87 views

To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 … (2n-1)\over 2.4.6…(2n)$

How can I show $\lim_{n \to \infty} a_n = 0$ $a_n = {1.3.5 ... (2n-1)\over 2.4.6...(2n)}$ I have shown that $a_n$ is monotonically decreasing. I thought to shown sequence is bounded from below ...
8
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10answers
170 views

How to evaluate $\lim_{n\to\infty}\sqrt[n]{\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}}$

Im tempted to say that the limit of this sequence is 1 because infinite root of infinite number is close to 1 but maybe Im mising here something? What will be inside the root? This is the sequence: ...
4
votes
3answers
53 views

Calculating the nth derivative of $\frac{x}{x+1}$

I was asked to calculate the nth derivative of $f(x) =\frac{x}{x+1}$. My solution: $$ f'(x) = (x+1)^{-2}$$ $$f''(x) = (-2)(x+1)^{-3}$$ $$f'''(x) = (-2)(-3)(x+1)^{-4}$$ $$f^{n}(x) = n!(x+1)^{-(n+1)} . ...
2
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3answers
36 views

factorial division when the bottom number is larger than the top number

I have a factorials problem to solve, and I do not know the method of solving it. I know how to do one number factorials (e.g. 5!, 15! etc...) and factorial division where the top number is larger ...
4
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2answers
83 views

Prove that $\sum_{n=1}^{\infty}\frac{1}{n(n+1)\cdots(n+a)}=\frac{1}{aa!}$ [duplicate]

$$\sum_{n=1}^{\infty}\frac{1}{n(n+1)\cdots(n+v)}=\frac{1}{vv!}$$ I am struggling to find a solution for this but no luck yet. How can I analyze it to get to second part?
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1answer
25 views

Factorial Divides Rising Power Proof Help

I'm trying to prove the following: $m^{\overline n} \equiv 0 \bmod n!$ Where $m^{\overline n} = m\left({m+1}\right)\left({m+2}\right)\ldots\left({m+n-1}\right)$, the product of $n$ successive ...
0
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0answers
55 views

Relation between Hyperfactorial, Superfactorial, Pascal's Triangle and Binomial Coefficient

I read here that the product of the elements in the $N^{th}$ row of Pascal's triangle is equal to $(n!)^{n+1}/(\prod_{k=1}^n k!)^2$. Let's call the product of elements in the $i^{th}$ row of Pascal's ...
1
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1answer
32 views

Rising factorial power

How the expression below can ve proved: $(a + b)^{\overline{n}} = \sum\limits_{j=0}^{n}C_n^j a^{\overline{n-j}}b^{\overline{j}}$ where $x^{\overline{n}}$ - is rising factorial power: ...
3
votes
1answer
103 views

How to prove that $\sum\limits_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)…(n+k)} = \frac{1}{kk!}$ for every $k\geqslant1$

Does anyone have any idea how to prove that $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)...(n+k)} = \frac{1}{kk!}$$
1
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4answers
87 views

Show that $(k!)^n$ divides $(kn)!$

Show that $(k!)^n$ divides $(kn)!$ I've tried it but without success. Any help would be great.
1
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0answers
38 views

Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$

We know that $\lim_{n\rightarrow \infty}\frac{n_1}{n}=p$ and $0\leq p\leq 1$. Based on this information I want to calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$. Any help? Note: ...